I think you left out the punchline in A2. Once I've got this gigungous matrix, then what? On Tue, Dec 4, 2018 at 7:37 PM Dan Asimov <dasimov@earthlink.net> wrote:
These are the questions from the recent Putnam Exam that seemed most interesting to me. I've changed some typography, and a word or two, for clarity. (I haven't tried to solve any yet. If anyone posts a solution, maybe include a spoiler warning for that problem number?)
—Dan
************************************************************************* ----- A2 Let S_1, S_2, ..., S_(2n−1) be the nonempty subsets of {1,2,...,} in some order, and let M be the (2n−1)×(2n−1) matrix whose (i,j)th entry m(i,j) = 0 if S_i ∩ S_j is empty; m(i,j) = 1 otherwise. -----
----- A5 Let f : R → R be an infinitely differentiable function satisfying f(0) = 0, f(1) = 1, and f(x) ≥ 0 for all x ∈ R. Show that there exist a positive integer n and a real number x such that the nth derivative of f is negative when evaluated at x:
f^(n)(x) < 0. -----
----- B2 Let n be a positive integer, and let
f_n(z) = n +(n−1)z + (n−2)z^2 +··· + z^(n−1).
Prove that f_n has no roots in the closed unit disk {z ∈ C: |z| ≤ 1}. -----
----- B4 Given a real number a, we define a sequence by x_0 = 1, x_1 = x_2 = a, and
x_(n+1) = 2 x_n x_(n−1) − x_(n−2) for n ≥ 2.
Prove that if x_n = 0 for some n, then the sequence is periodic. -----
----- B5 Let f = (f_1, f_2) be a function from R^2 to R^2 with continuous partial derivatives ∂f_i/∂x_j that are positive everywhere. Suppose that
(∂f_1/∂x_1)(∂f_2/∂x_2) - (1/4)(∂f_1/∂x_2 + ∂f_2/∂x_1)^2 > 0
everywhere. Prove that f is one-to-one. ----- *************************************************************************
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